Fourier Transforms of Lipschitz Functions on Compact Groups

Abstract

<p>If a function f is in LP(G), where 1 < p ≤ 2 and G is a locally compact abelian group, it is well-known that the Fourier transform f of f lies in L^q(r), where 1/p + 1/q = 1 and r is the dual group of G. This thesis is concerned with how this fact can be strengthened if it is known that f satisfies a Lipschitz condition. For certain kinds of compact groups (the circle and a-dimensional groups) we prove that if f is in Lip(α;p) then f lies in Lᵝ(r) for β > p/(p+αp-1), and a similar result holds for the n-dimensional torus. These results are generalizations and analogues of classical theorems of Bernstein and Titchmarsh about Fourier series and integrals. Furthermore we obtain more precise information for the case p = 2.</p>